A-Level Recurrence and general sequences Practice Questions

Free A-Level Recurrence and general sequences practice questions with full step-by-step worked solutions. Covers recurrence, sequences, series, limit. Practise exam-style problems and check your method.

recurrencesequencesserieslimitconstantsclassification
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
A sequence is defined by un+1=2un+3u_{n+1}=2 u_{n} + 3 with u1=1u_1=1. Find u2u_{2}.
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Worked solution

  1. Write down the recurrence and the first term

    un+1=2un+3,u1=1u_{n+1}=2 u_{n} + 3,\quad u_1=1

    Each term is generated from the one before it.

  2. Compute the term u_2

    u2=2×1+3=5u_{2}=2\times 1+3=5

    Apply the recurrence relation to the previous term.

  3. State the required term

    u2=5u_{2}=5

    This is the value that was asked for.

Answer
55
Question 2
2 markseasy
A sequence is defined by un+1=−1unu_{n+1}=-\dfrac{1}{u_{n}} with u1=2u_1=2. What is the period of the sequence?
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Worked solution

  1. Write down the recurrence and the first term

    un+1=−1un,u1=2u_{n+1}=-\dfrac{1}{u_{n}},\quad u_1=2

    The rule generates each term from the previous one.

  2. List the first few terms

    u1=2, u2=−12, u3=2, u4=−12u_{1}=2,\ u_{2}=- \frac{1}{2},\ u_{3}=2,\ u_{4}=- \frac{1}{2}

    Compute terms until the values start to repeat.

  3. State the period

    period=2\text{period}=2

    The sequence repeats every 2 terms.

Answer
22
Question 3
3 marksintermediate
A sequence is defined by un+1=11−unu_{n+1}=\dfrac{1}{1-u_{n}} with u1=4u_1=4. What is the period of the sequence?
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Worked solution

  1. Write down the recurrence and the first term

    un+1=11−un,u1=4u_{n+1}=\dfrac{1}{1-u_{n}},\quad u_1=4

    The rule generates each term from the previous one.

  2. Compute the term u_2

    u2=11−4=−13u_{2}=\dfrac{1}{1-4}=- \frac{1}{3}

    Apply the recurrence relation to the previous term.

  3. Compute the term u_3

    u3=11−(−13)=34u_{3}=\dfrac{1}{1-\left(- \frac{1}{3}\right)}=\frac{3}{4}

    Apply the recurrence relation to the previous term.

  4. Compute the term u_4

    u4=11−34=4u_{4}=\dfrac{1}{1-\frac{3}{4}}=4

    Apply the recurrence relation to the previous term.

  5. Identify the repeating block

    u4=u1 ⇒ period=3u_{4}=u_1\ \Rightarrow\ \text{period}=3

    The terms return to the start after 3 steps.

  6. State the period

    period=3\text{period}=3

    The sequence repeats every 3 terms.

Answer
33
Question 4
5 markshard
The sequence defined by un+1=−1unu_{n+1}=-\dfrac{1}{u_{n}} with u1=5u_1=5 is best described as:
Show worked solution

Worked solution

  1. Write down the recurrence and the first term

    un+1=−1un,u1=5u_{n+1}=-\dfrac{1}{u_{n}},\quad u_1=5

    The recurrence defines the whole sequence.

  2. Compute the term u_2

    u2=−15=−15u_{2}=-\dfrac{1}{5}=- \frac{1}{5}

    Apply the recurrence relation to the previous term.

  3. Compute the term u_3

    u3=−1(−15)=5u_{3}=-\dfrac{1}{\left(- \frac{1}{5}\right)}=5

    Apply the recurrence relation to the previous term.

  4. Compute the term u_4

    u4=−15=−15u_{4}=-\dfrac{1}{5}=- \frac{1}{5}

    Apply the recurrence relation to the previous term.

  5. Compute the term u_5

    u5=−1(−15)=5u_{5}=-\dfrac{1}{\left(- \frac{1}{5}\right)}=5

    Apply the recurrence relation to the previous term.

  6. Compute the term u_6

    u6=−15=−15u_{6}=-\dfrac{1}{5}=- \frac{1}{5}

    Apply the recurrence relation to the previous term.

  7. Compute the term u_7

    u7=−1(−15)=5u_{7}=-\dfrac{1}{\left(- \frac{1}{5}\right)}=5

    Apply the recurrence relation to the previous term.

  8. Compute the term u_8

    u8=−15=−15u_{8}=-\dfrac{1}{5}=- \frac{1}{5}

    Apply the recurrence relation to the previous term.

  9. Examine the behaviour of the terms

    u1=5, u2=−15, u3=5, u4=−15, u5=5u_{1}=5,\ u_{2}=- \frac{1}{5},\ u_{3}=5,\ u_{4}=- \frac{1}{5},\ u_{5}=5

    Looking at successive terms reveals the pattern.

  10. State the classification

    a periodic sequence\text{a periodic sequence}

    This describes the long-term behaviour of the sequence.

Answer
a periodic sequence\text{a periodic sequence}
Question 5
8 markschallenging
The sequence defined by un+1=−1unu_{n+1}=-\dfrac{1}{u_{n}} with u1=4u_1=4 is best described as:
Show worked solution

Worked solution

  1. Write down the recurrence and the first term

    un+1=−1un,u1=4u_{n+1}=-\dfrac{1}{u_{n}},\quad u_1=4

    The recurrence defines the whole sequence.

  2. Compute the term u_2

    u2=−14=−14u_{2}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  3. Compute the term u_3

    u3=−1(−14)=4u_{3}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  4. Compute the term u_4

    u4=−14=−14u_{4}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  5. Compute the term u_5

    u5=−1(−14)=4u_{5}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  6. Compute the term u_6

    u6=−14=−14u_{6}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  7. Compute the term u_7

    u7=−1(−14)=4u_{7}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  8. Compute the term u_8

    u8=−14=−14u_{8}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  9. Compute the term u_9

    u9=−1(−14)=4u_{9}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  10. Compute the term u_10

    u10=−14=−14u_{10}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  11. Compute the term u_11

    u11=−1(−14)=4u_{11}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  12. Compute the term u_12

    u12=−14=−14u_{12}=-\dfrac{1}{4}=- \frac{1}{4}

    Apply the recurrence relation to the previous term.

  13. Compute the term u_13

    u13=−1(−14)=4u_{13}=-\dfrac{1}{\left(- \frac{1}{4}\right)}=4

    Apply the recurrence relation to the previous term.

  14. Examine the behaviour of the terms

    u1=4, u2=−14, u3=4, u4=−14, u5=4u_{1}=4,\ u_{2}=- \frac{1}{4},\ u_{3}=4,\ u_{4}=- \frac{1}{4},\ u_{5}=4

    Looking at successive terms reveals the pattern.

  15. State the classification

    a periodic sequence\text{a periodic sequence}

    This describes the long-term behaviour of the sequence.

Answer
a periodic sequence\text{a periodic sequence}

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